Question

Use linear approximation, i.e. the tangent line, to approximate 3.76 as follows:
Let f(x)=x6. The equation of the tangent line to f(x) at x=4 can be written in the form y=mx+b
where m is:

and where b is:

Using this, we find our approximation for 3.76 is

Answer 1

Given $f(x)=x^6$

Tangent line to $f(m)$ of $x=4$

at $x = 4 : y = x^6; y = 4^6 \implies$ $(4, 4^6)$

Slope: $m(y) = y`= 6*5$

$m = 6*4^5$

$y - y_o = m(x-x_o)$

$y - 4^6 = 6*4^5[x-4]\\ \\y = x[6*4^5] - 24*4^5+46\\y=6*4^5[x] - 4^5*(24-4)$

$y = 6*4^5[x] - 20*4^5$

So compaing with y = m*x+c:

m = $6(4)^5;$ c = $-20(4)^5$

To estimate value of $3.7^6$ using linear approximation:

f(x) = f(a)+f`(a)(x-a)

so $f(3.7) = f(4) + f`(4)(3.7-4) = 4^6+6*4^5*(-0.3)$

$= 4^6 - 6*(0.3)*(4)^5 = 2252.8$

So, linear approximation to estimate $3.7^6$ is 2252.8